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The Cs 6S-7S-6P3/2 forbidden three-level system : analytical description of the inhibited fluorescence and
optical rotation spectra
M. A. Bouchiat, J. Guéna, Ph. Jacquier, M. Lintz, L. Pottier
To cite this version:
M. A. Bouchiat, J. Guéna, Ph. Jacquier, M. Lintz, L. Pottier. The Cs 6S-7S-6P3/2 forbidden three-
level system : analytical description of the inhibited fluorescence and optical rotation spectra. Journal
de Physique, 1989, 50 (2), pp.157-199. �10.1051/jphys:01989005002015700�. �jpa-00210910�
The Cs 6S-7S-6P3/2 forbidden three-level system : analytical description of the inhibited fluorescence and optical rotation spectra
M. A. Bouchiat, J. Guéna, Ph. Jacquier, M. Lintz and L. Pottier
Laboratoire de Spectroscopie Hertzienne (*) de l’Ecole Normale Supérieure, 24 rue Lhomond,
F-75231 Paris Cedex 05, France
(Reçu le 27 mai 1988, accepté le 14 septembre 1988)
Résumé.
-En vue d’améliorer la précision des mesures de violation de parité dans le césium,
nous avons étudié le « système interdit à trois niveaux » 6S-7S-6P3/2, dans lequel un premier laser
excite la transition interdite 6S-7S pendant qu’un deuxième faisceau, colinéaire au premier, sonde
les atomes excités dans le niveau 7S. Dans cet article, nous présentons un calcul analytique de la
fluorescence du niveau 7S, ainsi que de l’amplification du faisceau sonde, en fonction des
fréquences des lasers. Nous prenons en compte les collisions dans le niveau de résonance
6P3/2, ainsi que la multiplicité des niveaux et les polarisations des lasers. L’accord quantitatif avec l’expérience est satisfaisant ; les spectres sans effet Doppler sont correctement décrits. Nous obtenons les amortissements de la cohérence 7S-6P3/2 et de la population emprisonnée dans 6P3/2. Nous présentons de nouveaux procédés de détection de l’orientation du niveau 7S, qui
doivent pouvoir être étendus directement à la détection d’un alignement.
Abstract. - With a view to improved parity violation measurements in Cs, we have considered the 6S-7S-6P3/2 « forbidden three-level system », in which one laser excites the forbidden 6S-7S transition while a second, colinear laser probes the excited 7S atoms. This paper presents an analytical calculation of the 7S ~ 6P ½ fluorescence intensity and of the probe amplification as
functions of the laser frequencies. The collisional processes in the 6P3/2 resonance level, as well as
the level multiplicity and the laser polarizations are taken into account. The quantitative
agreement with experiment is good ; in particular, the observed sub-Doppler structures are
correctly described. The damping rates of the 7S-6P3/2 cohérence and of the trapped 6P3/2 atoms are extracted. New détection schemes for detecting a 7S orientation, with direct
possible extension to an alignment, are demonstrated.
Classification
Physics Abstracts
32.70
-32.80
-33.35
-35.10W
1. Introduction.
In recent years, atomic physics parity violation (PV) experiments [1] have contributed to the
knowledge of weak interaction processes. Experiments in which a laser beam excites a
forbidden transition have succeeded in measuring the so-called « weak charge » of the cesium
(*) Associé au CNRS et à l’université P. et M. Curie.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005002015700
nucleus. This is the additive parameter, analogous to the electric charge, that characterizes the Zo-nucleus vector coupling [2, 3]. The result of the measurements [4, 5] matches the
prediction of the Standard Model of weak, strong, and electromagnetic interactions.
However, the weak charge is a fundamental parameter. Its value is sensitive to higher order
corrections that involve virtual emission or absorption of particles. Measuring the nuclear weak charge of cesium a factor ten more accurately gives access to these corrections, and dramatically changes the significance of the comparison between experiment and theory.
In order to improve the statistical accuracy, the detection of the excited atoms has been reconsidered [6]. Rather than analysing fluorescence light, monitoring a transmitted probe
beam allows in principle the detection of all excited atoms, in a single direction and a single transition, thus leading to better detection efficiency. This is the direct motivation for the
present work, which deals with the interaction of a vapour with two laser beams : one excites the forbidden 6S-7S transition ; the other probes the final state of the transition. Ultimately,
since parity violation takes place in the excitation process, the second laser will serve to detect the resulting breaking of mirror symmetry in the final state.
The present stage is preliminary. It aims at exploring the potentialities of this method and at fully understanding the parity conserving processes in this situation. Since the excitation laser connects two states of the same parity, the distribution of parities in this three-level, two-laser system is quite unusual. For that reason we call it a « forbidden three-level system ».
In this paper we present a theoretical approach to the forbidden three-level system formed by Cs vapour in the presence of two resonant single-mode cw lasers, one of which excites the 6S-7S forbidden transition, while the other probes the 7S-6P3i2 transition. Agreement with experimental data is illustrated, and damping rates of the system are obtained. Moreover it is shown that even PV measurements by fluorescence detection can benefit from the use of the
7S-6P3/2 probe.
1.1 THE ATOMIC TRANSITIONS.
-As in the previous PV experiments in Cs, a cw laser beam
excites the 6S-7S highly forbidden Ml transition (oscillator strength
=10- 15) in the presence of a transverse electric field Es (a few hundred V/cm ; oscillator strength up to
10-1°). The 6S and 7S hfs splittings (level diagram in Fig. la) are much greater than the Doppler width, and only one component is excited. The Stark-induced El transition is described by the scalar and vector polarizabilities a and /3 (j/3 1 ===== 1 a 1/10) [2].
In this paper we first consider an excitation laser of linear polarization £e//Es ; 13 is not involved, and the only effect of the laser is to bring population in the 7S state. There is no
orientation. Later on, when the laser is polarized circularly, we shall deal with a 7S orientation proportional to {3/ a.
The allowed 7S-6P 3/2 transition has a large oscillator strength : = 0.44. The hfs in
6P3/2 is only
=200 MHz and cannot be resolved in Doppler-broadened spectroscopy. The 7S --> 6P spontaneous emission populates the 6P3/2 and 6P 1/2 levels with branching ratios of roughly 2/3 and 1/3. The Norcross model [7] predicts 0.649 and 0.351. But to our knowledge
there is up to now no precise empirical determination.
1.2 THE FORBIDDEN THREE-LEVEL SYSTEM. - The basic concept of a three-level system driven by two resonant lasers has attracted a great deal of experimental and theoretical efforts in the past twenty years (an excellent list of references can be found in [8]). In spite of the apparent similarity, the forbidden three-level system differs from the systems considered in usual three-level spectroscopy. This results from the unusual distribution of the parities
among the system. One obvious difference lies in the very small excitation rate of the
forbidden transition, which requires only a lowest order treatment. Furthermore, while the
coherent process (here, direct 6S --> 6P3/2 two-photon excitation) dominates in usual three-
Fig. 1.
-a) Relevant energy levels of the Cs atom with their hyperfine structure (not to scale. Natural cesium contains only 133CS, with I = 7/2). b) Radiative and collisional couplings between the levels.
Straight lines indicate the laser couplings ; wavy arrows indicate spontaneous emission. The outer box delimits the two-level system mentioned in § 1.2. Precise definitions of the transition and damping rates
are given in section 1.2.
level systems, the forbidden three-level system essentially undergoes incoherent stepwise
processes (6S-->7S absorption, then 7S ---> 6P3/2 induced emission). The efficiency ratio of stepwise to coherent processes is known to be comparable to the ratio F7S/Fr of the damping
rates of the intermediate (7S) level and of the two-photon (6S-6P3/2) coherence [9]. The
collisional broadening of the 6S-6P resonance lines [10,11] is known to be very efficient at the Cs densities of interest here (several x 1014 at/CM3) while the 7S level is nearly unaffected by
Cs-Cs collisions [12]. As a consequence, the relative contribution of the coherent process is
significantly reduced as soon as collisional broadening takes place. In addition, a quantitative
calculation shows that the coherent contribution is further reduced by the energy ratio of the forbidden transition to the 7S-6P3/2 transition (= 2.72). Therefore the contribution of the coherent two-photon process is smaller even at Cs densities low enough for no significant
collisional broadening to take place. As a result, the vapour under the action of the two laser beams, behaves merely as a two-level system (the 7S-6P3/2 system, outer box in Fig.1b) coupled to the probe beam. Because of resonance radiation trapping, population escapes this two-level system by 7S --. 6P 1i2 spontaneous decay more easily than by emission of
6P3/2 - 6S resonance photons.
1.3 THE MODEL. - In the experiments, both the 7S --> 6Pl/2 fluorescence intensity and the intensity and polarization of the transmitted probe beam have been detected. Special
attention has been paid to contributions proportional to the helicity of the excitation beam.
All observed signals are calculated below. The hyperfine structure and the collisional redistribution in 6P3/2 are taken into account. The following assumptions are made :
i) The two cw single-mode lasers, whose frequency jitters do not contribute to the widths
by more than 1 MHz each, are considered to be monochromatic. Their intensity is assumed to be uniform over the volume of vapour.
ii) Each population or coherence is calculated at the lowest non-zero order of the forbidden transition.
iii) The coherent two-photon effect is neglected. The validity of this assumption is checked
in appendix A.
iv) The collisional redistribution in the 6P3/2 resonance level is described by a collision
model where a 6P3/2 atom transfers its excitation to a 6S atom while their velocities remain
unchanged. After the collision, the distribution of the velocity of the new 6P3/2 atom is
therefore assumed to be thermal. This « strong collision » model is a good description of
resonance collisions as well as of emission and reabsorption of a resonance photon. Since the
collision time is smaller than the inverse 6P3/2 hfs splitting, the hf momentum is also assumed to be redistributed over all values (F’
=2 to 5) with probabilities proportional to the multiplicities 2 F’ + 1.
v) The effect of Cs-Cs collisions on 7S atoms is considered to be negligible.
vi) The collisional transfer between 6P1/2 and 6P3/2 is neglected (see discussion in § 3.5.2).
vii) Due to the complexity that results from large angular momenta, one has to omit the higher order tensors (K > 3). This approximation restricts to low saturation values (s 1) the
domain of quantitative validity of the solution, but allows an analytical treatment to all orders.
viii) The width of the probe transition is considered to be large as compared to the width of the 7S population source, but small as compared to the Doppler width. As long as the
saturation broadening remains moderate, the integrations over the velocity distribution then
simplify and an analytical solution is obtained.
The good agreement between experimental and theoretical data allows us to extract from theoretical fits the damping rates of the 7S-6P3/2 system, and the value of the 7S orientation created by a circularly polarized excitation laser.
2. The atomic évolution in the présence of the two cw resonant lasers.
2.1 FORMULATION OF THE PROBLEM. - We shall develop a semi-classical treatment of the
problem. The main notations are illustrated in figure 1b.
2.1.1 The lasers fields.
-The cw lasers are described by classical fields :
where j = e represents the (green) laser exciting one hf component of the 6S-7S transition, and j = d represents the (I.R.) detection laser probing the 7S-6P3/2 transition. The 6S-->7S transition is highly forbidden, even in the presence of the Stark field. As a result, 7S atomic densities in the experiments remain small. Therefore we neglect in this first stage any alteration in the field amplitude, polarization and phase caused by propagation ; this problem
will be treated in section 5.
For an atom of velocity v, the frequencies of the fields in the atom’s frame are shifted by longitudinal Doppler effect :
We have assumed that the two beams propagate in exactly the same direction of unit vector
k. We define the quantity
which is the Doppler shift for the probe beam. From now on, it will be convenient to call
« velocity » the quantity v instead of v. So, for an atom of velocity v, the time dependence of
the laser fields reads
The frequency of the probe beam is swept over frequency intervals of at most 3 GHz, so that
in practice we/wd never deviates from the approximate value 2.72.
2.1.2 The cesium vapour.
-Each velocity class of the atoms in the vapour is described (in
the interaction representation) by a density matrix p ( v ). The number of Cs atoms per unit volume whose velocity lies between v and v + d v is tr {p ( JI )} dv . Finally only the subspace spanned by the 6S1i2F, 7S1/2F, and 6P3/2F’ levels (1) is considered.
2.1.2.1 The populations and their relaxation.
-Using the notation l?(aF)ae Lia, Fm>
m
a , Fm 1 for the projector over the «F hfs level, we define the density matrices and the populations of the involved levels by :
Since the fraction of excited atoms is always very small, the ground state remains thermally populated and we can write :
Here I = 7/2 is the nuclear spin of 133CS ; ncs is the atomic Cs density and
is the normalized Boltzmann velocity distribution; nD == Cù7S-6P312 .J kT / Mes C2 is the
Doppler half-width at 1/ à (
=21T x 110 MHz for the temperatures of interest here).
The only relaxation process in the 7S state is the radiative 7S ---> 6P decay :
where 1/ r 7S
=48 ns is the radiative lifetime of the 7S level [12]. The same relation holds for n7FCv ).
For the resonant 6P3/2 states, on the contrary, resonant collisions as well as resonance
radiation trapping have to be taken into account. The equation that governs the evolution of the population under relaxation contains several contributions :
The first term describes the 7S --> 6P3/2 spontaneous emission. For the branching ratios of
the 7S --> 6P3/2 and 7S --> 6P,/2 fluorescence we use the values 2/3 and 1/3 (which take no
account of spin-orbit effects). The relative oscillator strengths FI are defined so that
2: C FI F = 1 (App. B).
F’
(1) The hyperfine momentum is indicated by F (= 3 or 4) in the 6S and 7S levels, and by
F’ (= 2, 3, 4, or 5) in the 6P3/2 levels.
The second term describes the global decay of the 6p3/2 population. Because of radiation
trapping, the rate r p is much less than the radiative decay rate of an isolated 6P3/2 atom [13]
(typically by two orders of magnitude, as shown in our data analysis, given below).
The third term of equation (5), in which
is the total 6P3/2 population, and
the relative degeneracy of the 6P3/2F’ level, describes the redistribution of the 6P3/2 atoms over
all thermal velocities and over all hfs levels. One can easily check that the third term does not
give rise to any global decay of the 6P3/2 population. On the other hand, a velocity class
relaxes with a rate Tp + F col’.
while the 7S-6P3/2 coherences
are damped
with a rate y which reflects both radiative and collisional processes.
2.2 THE MASTER EQUATION AND THE STEPWISE APPROXIMATION.
-For the class of
velocity v, the evolution of the density matrix is given by
where X(t, v) is the Hamiltonian for the atom-field coupling in the interaction representation :
(D is the electric dipole operator, X o the Hamiltonian of an atom in the Stark field).
Applying the rotating wave approximation, we neglect the non-resonant terms in the matrix elements of H(t, v), that is, the coupling between atomic states for which the detuning is much
greater than the Doppler width (all other widths are smaller than f2D), We obtain
where Ve = {úJe - (E7sF - E6sF)/h} /2.72 is the velocity of the atoms resonant with the green
(excitation) laser, and wFF = (E 7SF - E6P 3/2, F’ )/h. Here we assume that the two lasers are
resonant for the same 7SF level (the opposite case will be studied in § 3.3) and that the green laser drives a àF
=0 transition.
As indicated in section 1, we shall neglect the 6S-6P two-photon coherence (except in appendix A where the validity of the approximation is checked). The first stage of the step-
by-step excitation involves the 6S -+ 7S transition. In view of its high forbiddenness (oscillator
strength _ 10- 9 for electric fields Es :5 1000 V/cm), a lowest order calculation is fully appropriate. The green laser is represented by a source term for the 7SF population. This
term is :
where n6F(v)=ncsf(v)(2F+1)/2(2I+1). The transition dipole matrix element d is defined as d - (6F, Fm 1 9). êe 7S, Fm) where 1 , Fm> denotes the states in the presence of the Stark field. We assume Ëe // Es, so that d involves only the scalar polarizability :
d
=« ( Es . Ëe 1
=a Es ; the vector polarizability 8 plays no rôle. (The case of circular polarization will be treated in section 4.) The quantity d is then independent of m. We do not give the details of the calculation leading to equation (8a) since it is very similar to the calculation leading to the coupling with the probe laser (Eq. (14)).
The source term turns out to be the product of a sharp lorentzian shape and a much broader
gaussian shape f (v). To a good accuracy, we may replace v by ve in the gaussian n6F( v) factor and rewrite equation (8a) as
where the total excitation rate A is a constant and g ( v ) is a normalized lorentzian function.
Now we just have to solve the master equation for the density matrix restricted to the 7S and 6P3/2 levels. Defining OFF, =- P(7SF) S) ’ Êd P(6P3/2 F’), this equation becomes
2.3 THE ATOMIC DENSITY MATRIX IN STEADY STATE.
-Let us derive, from equation (9), the equations governing the evolution of the coherences and the populations :
We are interested in the steady state of the vapour, for which the populations are constant in
time and the coherences PFF’ (v) oscillate like ei(WFF’ - (J)d + V) t Any other term would bring
oscillatory terms in the right hand side of equations (11) and (12). Then, from equation (10),
one easily gets an expression for the coherences :
which can be carried back in equations (11) and (12). Finally the steady-state equations for
the 7S and 6P 3/2 populations are
where we have introduced the notation àj, - (úJ FF’ - wd + v )/ y.
In equations (14), (15), it appears that then steady-state populations are not only coupled to
the populations, but also to other elements of the density matrices of the 7S and
6P3/2 levels. Writing a detailed expression for the operators DFF’ DFF’ (which acts on the 7SF level) and ’D:FI Dpp (which acts on the 6P3/2 F’ level) will clarify the couplings between populations, orientations, alignments, etc...
2.4 THE COUPLINGS OF POPULATION TO ORIENTATION AND ALIGNMENT. - Let us consider
F F
two tensor operators ...g! and...g t of rank 2, such that for any two vectors u, v, the following
F’ F’
relations hold :
F
Then one has Dpp’ DFF = ed ’c fI. In Appendix B we have performed the decomposition
F’
of ..g and É î on the basis of the irreducible tensor operators p(O), p(1), and
p(2), of ranks 0, 1, and 2, defined by
The decomposition of ’G- 1 reads :
The coefficients C, a ~, b 1 (and the corresponding coefficients a t and b t in the
decomposition of r 1 ) are given in § 2 of Appendix B. This results in a convenient expression
for the operator DFF’ DFF’ :
According to equations (14), (15) the observable quantities coupled to the populations are
Tr {DFF’ DFF,P7F(v)) and the analogous quantity involving PPF’(V)’ Using equation (20) it is reexpressed as a combination of three terms : i) Tr { F(F3+ 1) P7F( JI)} oc n7F( JI) : the probe
beam couples the populations, independent of its polarization. ii) Tr {F. (êd A Ê!) P7F( v)} :
if the probe is circularly or elliptically polarized, it couples the populations to the orientation.
iii)
the alignment along êd is also coupled to the populations.
In some of our experiments, the probe beam was circularly polarized [14] ; the correspond- ing calculation is developed in section 4. Here, with a view to interpreting all spectra recorded with a linearly polarized probe beam, the population-orientation coupling will be omitted.
Although the green laser does not create any alignment in the geometrical configuration
considered here, yet the alignment may not be a priori neglected. The reason is that the probe beam, when it saturates the 7S-6P3/2 transition, creates an alignment in the 7S level (2). The alignment itself is coupled to the rank-4 tensor and the exact solution would require to take
into account all the even rank tensors.
All the spectra from which quantitative values have been extracted were obtained with a
weakly saturating probe beam. As shown more explicitly in section 4, it is then justified to
restrict ourselves to an approximate solution which omits the alignment and results in
important simplifications. Actually, as will appear in section 4, we have been able to take into
account the laser-induced alignment (but not the higher-order tensors). The corresponding
correction to the spectral shape is less than 10 % for the saturation levels of the experiments (see Fig. 2). No qualitative modification results, even for strong saturation.
Neglecting the alignment we write
and
(2) So does it in the 6P 3/2 level as well ; but all tensor quantities in the 6P levels are quickly washed off
by resonant collisions.
Fig. 2.
-Fractional increase of fluorescence caused by the probe beam, versus probe frequency.
Example spectra calculated without (solid line) or with (dashed line) the alignment-induced corrections.
Values of the parameters : s
=0.8 ; y/27r = 18 MHz ; rcoll/21T
=36 MHz ; rp/21T = 100 kHz.
This gives
where
To conclude this section, we can now write the steady-state equations for the 7SF and the
four 6P3/2 F’ populations :
where we have introduced
The quantity f2 R = 1 Ed (7811211 9) ,,6P3/2) III J61 I is the « Rabi pulsation » for the 7S-
6P3/2 transition. Note that the Rabi pulsation for a particular 7SF ---> 6P3/2 F’ transition is
actually nR CF,. Since CF, ranges from 0.10 to 0.61, we expect different saturation levels from one hfs component to another. (On the other hand, neglecting the alignment, we are led
to neglect the variations of the Rabi pulsations associated with different Zeeman components
belonging to the same hfs component).
3. Calculation of the « inhibited fluorescence » spectra.
Starting from equations (23) and (24) we are now going to calculate the total 7S population in
order to obtain the 7S --+> 6P,/2 fluorescence in the presence of the probe beam.
3.1 CALCULATION AT THE LOWEST ORDER IN THE PROBE INTENSITY.
-In absence of the
probe beam (R’ (v)
=0), the expression nj$( v ) = Ag (v)/r7s for the 7S population can be
inserted in equation (24) to obtain :
Next we get the modification of the 7S population to first order in RFF(v) :
and the resulting increase of the 75 - 6P 1/2 fluorescence :
This expression involves two different integrals. Instead of a numerical study (the first integral
cannot be expressed analytically) we shall use an approximation. Taking advantage of the hierarchy between the half-widths of g (v), RFF(v), and f ( v )
we consider that RFF(v) is broad as compared to g(v), but sharp as compared to f(v ). Then the integrations are carried out easily by treating the sharper factor as a ô-
function :
In addition to the usual saturation parameter
we shall introduce two dimensionless parameters :
and the « contrast » parameter
Finally, at the lowest order, we write the increase of fluorescence
where the summation 1 reduces to three terms (CF,
=0 for F’
=F ± 2), corresponding to
F’
the three dipole-allowed transitions from the populated 7SF level to the 6P3/2F’ levels.
Interpretation.
-The fluorescence increase (as a function of wa) is the sum of sharp, negative
lorentzian terms, and of broad, positive, gaussian-shaped terms. The sharp holes reflects
« inhibition » (actually reduction) of the fluorescence intensity due to the competition of
stimulated emission (« inhibited fluorescence » holes). The broad contribution corresponds to
the reexcitation of the population trapped in the 6P3/2 resonance level. These reexcited atoms
get a new chance to be observed in spontaneous decay.
The holes are not exactly centered on the frequencies wFF/2ir of the 7SF ---> 6P3/2F’
transitions, but are all Doppler-shifted by a quantity ve/2r proportional to the velocity of the
atoms excited in the forbidden transition. The hole depth is affected by a factor
1-2 KCF 1£F /3 : this accounts for the presence of velocity-selected atoms in the 6P3/2 levels caused by spontaneous decay of the 7S atoms. Since K 1 (Eq. (26b)), this factor
actually remains close to unity, except at very low Cs pressure.
We define the « contrast » of the spectra as the ratio of the hole depth to the height of the corresponding Doppler line. Omitting the factor rcoll/ (r p + rcoll), very close to one since
rp « rcoll, the contrast is C/It F’WF’ F e x 32/(2 F + 1). It does not depend on F’.
3.2 NON-PERTURBATIVE SOLUTION.
-First subtracting equation (24) (summed over F’)
from equation (23), we get
Then, defining the total 7S population we obtain the relationship (3) :
which expresses the balance between the numbers of atoms excited to the 7S level (source
(3) Throughout sections 3 to 6, we make frequent use of the relations
term A) and of atoms which leave the system, either by 7S --> 6P,/2 fluorescence (jTys fi7/3), or by 6P3/2 --> 6S decay (r p nP). We can now calculate the 6P3/2 population.
Inserting in equation (24) the expression of n7F ( v ) obtained from equation (28), we obtain
an expression for nPF(v) as a function of s nPF’(V) and nP, which leads to
where
and
Introducing the dimensionless quantities (functions of the laser frequencies -we and
we obtain the total 6P3/2 population :
In the absence of the probe beam, this result is consistent with the zero-order value
since, as we shall see later, F, 9 oc s when s - 0.
Finally, equations (29) and (32) allow one to calculate the total 7S population, and the
fractional increase of the fluorescence intensity caused by the IR laser :
Let us recall that, if one forgets the two-photon processes, equation (33) is exact for an unpolarized probe beam.
We have obtained a very simple form for the fluorescence increase. But the quantities Y
and 19 are given by rather intricate definitions. A numerical interpretation would not allow an
easy physical interpretation. On the contrary, an approximate but analytical expression
appears to be very useful.
Approximate expressions for 19 and :F. - The same kind of approximations as for the lowest
order calculation (§3.1) will be used for the velocity integrals. The sharp function 9 (v) is replaced by 5( v - v e) . This remains valid at high saturation (see App. A). Then we
have simply
It is useful to separate the w d-independent part in I(ve). This leads to :
where à§, = (úJFF’ - úJd + Ve)!Y. One finally gets
where we have introduced the lorentzian-shaped function :
To obtain an approximate expression for [f, we insert equations (30) in equation (31b), and
write
The integral involves the product of f(v), the thermal distribution, by a function corresponding to three resonances of much smaller widths. We note that the saturation
broaderiing of the Lorentz functions (width 2 y .JI + IL:’ CF, Ks) appears only at strong
saturation (s > 1), since
In addition, saturation induced overlapping of the Lorentz functions (for s > 1) makes the
denominator in equation (37) nearly constant between the two extreme hfs components. Thus only the high (resp. low) frequency wing of the F’
=F - 1 (resp. F’
=F + 1) component brings additional broadening.
We now proceed in the « Doppler limit », that is, supposing that f ( v) can be taken as
constant over the width of a resonance. This approximation is quantitatively valid for low or
middle saturations. For strong saturations, it still provides a very useful description of the
behaviour of the fluorescence signal. In the Doppler limit, we have
where
(We have assumed that the resonances are well resolved.)
Now, remembering that we finally obtain
The weights ’TTF" close to 1 at low saturation, embody all saturation effects ; they are given by
Interpretation. - The fluorescence increase induced by the probe beam (Eq. (33)) involves
the two quantifies 9 and 37, whose respective significances are clear :
i) 9 (Eq. (31a)) is related to the sharp function g (v) which describes the 7S velocity
distribution in the absence of the probe laser (Eq. (8b)). The presence of 9 in Jf/JlO)
describes the decrease of the fluorescence due to the 7S --> 6P3/2 stimulated emission (minus sign in (Eq. (33)).
ii) 37 (Eq. (31b)) is related to the thermal distribution f(v). Equation (39) shows that Y is
proportional to 1/C, that is, to the lifetime 1 /rp of the trapped 6P312 population (Eq. (26c)).
The term Y in equation (33) describes the reexcitation of 6P3/2 atoms, which results in an
increase of the fluorescence.
As opposed to the lowest-order expression (Eq. (27)), the non perturbative solution is not the sum of three independent terms for the three allowed transitions, and this is due to
saturation. Saturation takes place in g, in F, and in the denominator (1 + :F/2).
In g, saturation appears in two different manners. From equation (35), when s£ --. oo at
strong saturation, then g -->. 1. In this case - 19 = - 1 means that the fluorescence is totally
inhibited : stimulated emission has induced all 7S atoms to decay to the 6P3/2 level. But stimulated emission cannot be that efficient if the 6P3/2 atoms are not quickly quenched or
redistributed by collisions. This is the reason of the term £F’ F’ Ks -5 rcs) in the denominator of equation (36), which prevents sf from taking large values if K - r 7S/ (r p + F"11) is not
very small compared to unity.
While g remains always smaller than 1, 37 grows to infinity as S1/2. Hence from equation (33)
the upper limit of the fluorescence increase AJf/Jf is 2
-or more precisely the branching ratio r -+3/2/ r -+ 1/2 of the 7S -. 6P fluorescence. The interpretation is that when the 6P3/2 --. 7S
reexcitation rate is high, the only possibility for the atoms to leave the 7S-6P3/2 system is the 7S -. 6Pl,z fluorescence, which is then multiplied by a factor 1 + (r -+ 3/21 r -+ 1/2 )
-3.
One notes that the term 19 in equation (33) is also affected by the denominator
(1 + F/2), which means that the depth of the inhibited fluorescence holes decreases at very strong saturation. The trapped 6P 3/2 atoms can be easily reexcited to the 7S state and this
reduces the efficiency of the induced 7S -. 6P3/2 emission when the probe field is saturating.
As will be seen in § 3.4 , this behaviour has been observed experimentally, in good agreement with the prediction.
As an example of the results of this calculation, we show a theoretical spectrum for
A34Wd)13f obtained from equations (33), (35), (36), (39) and (40) (solid line of Fig. 2). The
parameters s,k,y,C have been assigned values corresponding to typical experimental
conditions. The dotted line displays the spectrum obtained when the 7S alignment is taken
into account by the same method as in section 4.
3.3 PROBE FIELD RESONANT WITH THE UNPOPULATED 7S hfs LEVEL.
-We now consider the
case of a green laser exciting the 7S, F, hfs level, and of a detection laser probing the 6P3/2 F’ H 7S, Fd transitions, with Fe =1= Fd. The hfs splitting in 7S is 2.2 GHz ; so there is no overlap between the two hfs levels. The master equations are similar to equations (23)-(24).
The only differences are the following : i) there is no source term for the 7S, Fd population (Ag(v) in Eq. (23) disappears) ; and ii) in the spontaneous 75 - 6P emission term of
equation (24), r 7S C:, n7P( v) is replaced by {CFe Ag( v) + r 7S C: n7P( v) } .
The method that led to equation (33) now leads to
where Y is defined as before (Eq. (31b) with Fd replacing F), and
(là’F’ =- ( - F, F’ - W d + ve )/ y) replaces equation (35). Instead of the previous three (inhibited fluorescence) holes (- 19 in Eq. (33)), the term + g now gives rise to two positive bumps (9’
involves the product C Fe C Fa which is zero unless F’
=3 or 4). The two bumps indicate the presence, in two of the 6P3/2 hfs levels coupled to the probe field, of a velocity-selected population resulting from the spontaneous decay of the 7S atoms (Fig. 3a). This velocity-
selected population is strongly damped by resonant collisions ; so the two corresponding bumps are expected to be nearly washed off by collisions, except at very low Cs pressure. A
typical low pressure spectrum is given in figure 3b.
3.4 COMPARISON WITH EXPERIMENT. - The comparison between theory and experiment
has been given in a previous paper [15]. We will recall it briefly here.
Numerous inhibited fluorescence spectra have been recorded at relatively low saturations.
All were found in good agreement with the calculated spectra. Only one slight discrepancy
has been detected on the wings of the spectra : the calculated fluorescence increase is slightly
less than the experimental one (see for instance Fig. 5 in [15]). This is connected with the
Doppler limit in the calculation of Y.
The agreement remained good in quite different experimental conditions :
i) Excitation selecting a class of atoms of non-zero velocity ( ve 0 in Eq. (34)). In this
case the holes turn out to be shifted by the expected amount with respect to the Doppler profile (Fig. 4 of [15]).
ii) Cs density varied from 5 x 1012 to 6.4 x 1014 at/cm3. The strong collisional broadening
of the 7S-6P3/2 coherence does not spoil the agreement between theory and experiment (Fig. 6
of [15]) ; neither does the modification of the trapping of the 6P3/2 population.
iii) Probe field resonant with the unpopulated level (Fe= Fd). The experimental spectrum
(Fig. 3c) displays the two expected peaks, just like the spectrum calculated in 3.3 (Fig. 3b).
We formed the ratios between the heights of the two bumps and the depths of the corresponding inhibited fluorescence holes (when Fe
=Fd), which are, up to an angular
momentum factor, nothing but the ratio K
=r7S/(rp + rcoll). The low pressure value thus obtained for TP + rcoll is close to the 6P3/2 radiative relaxation rate, as expected when Cs-Cs
collisions are negligible.
Fig. 3. - Probe laser resonant with the unpopulated level a) Level diagram illustrating the case where
the green laser excites the 7S, F
=3 level while the IR laser probes the 7S, F
=4 - 6P3/2 F’ transitions.
b) Theoretical spectrum for the fluorescence increase at low Cs density. c) Experimental spectrum. Cs density 5 x 1012 at/cm3 ; IR intensity 13 mW/cm2 ; electric field Es
=1 800 V/cm.
Having firmly established the quantitative validity of the calculated spectral shape for low
or moderate saturation, we also tested the validity at strong saturation. The most salient features predicted for s --> oo are i) the equality between the maximum value for the fluorescence increase and the branching ratio r -+ 3/2/ r -+ l2 of the 7S fluorescence (theoretical
value 1.85 [7]) ; ii) the decrease of the hole depth ; iii) the radiative broadening of the holes.
At first sight the maximum value of Jf/jf is a very direct measure of the branching ratio.
However, the above model assumes the 6P3/2 and 6P,/2 levels to be uncoupled. Actually,
collisions with ground state atoms induce excitation transfers between them [16]. When the 6P3/2 level is depopulated by the strongly saturating probe laser, transfer takes place from 6P1/2 to 6P3/2. This gives some chance to an atom that has decayed to 6P1/2 to be cycled once or
more. Present lack of information about the transfer cross section at the temperature of our
measurements prevents quantitative conclusion. But we believe that this explains why we
have observed, at strong saturation, values of Jf/Jf larger than r -+ 3/2/ r -+ 1/2 (for example
= 2.1 for ncs
=2 x 1014 cm- 3 and = 2.4 for ncs
=6 x 1014 cm- 3).
The predicted decrease of the hole depth when s - oo has been evidenced owing to a
detection procedure particularly sensitive to non-linearities. The probe intensity (and
consequently the saturation) was modulated sinusoidally between zero and a strongly saturating intensity, and the resulting modulation in the fluorescence intensity was detected at
the same frequency. Qualitatively, the information obtained this way is the difference between the fluorescence intensities for high and low saturations, i.e. roughly the derivative d3flds. In the linear regime (modulation amplitude 6s « 1), this procedure gives of course the
same spectra as the bare fluorescence increase. But as soon as saturation effects become
significant, the non-linear s-dependence of the hole depth manifests itself in a reversal of the holes (Fig. 4a). At very large modulation amplitudes, the reversal is such that the holes are
turned into sharp peaks (Fig. 4b), indicating that the holes are deeper for s = 1 than for s > 1. The agreement with the corresponding calculated spectra demonstrates the qualitative validity of our theoretical model at strong saturation.
Fig. 4. - Lock-in detection of saturation induced non-linear effects. The probe intensity is modulated
sinusoidally. The resulting modulation in the fluorescence intensity is plotted as a function of the probe frequency (Cs density 5 x 1012 at/cm3) . The theoretical spectra are obtained by numerical calculation of the integration performed by the lock-in detection.
3.5 QUANTITATIVE RESULTS OBTAINED WITH THE HELP OF THE MODEL. The most
important for us was to study the modification of y and T P with the cesium density
ncs. This was achieved by fitting, to each experimental record, the theoretical spectrum obtained by adjusting the damping rate y of the 7S-6P3/2 coherence, the lifetime
TP of the trapped 6P3,2 population, the saturation parameter s, and the detuning
Ve of the excitation laser. This quantitative analysis includes only spectra taken with a non-
saturating probe (s = 1).
3.5.1 Evolution o f y with ncs. - The parameter y plays an important rôle in determining the optimum conditions for the future parity-violation experiment [6]. While the collisional
broadening of the resonance lines was known [10, 11], that of the 7S-6P3/2 transition was not.
Figure 5 shows the plot of y versus ncs obtained from our measurements. The solid line is a
best fit to the experimental points, and gives
with a slope uncertainty mainly associated with the Cs density calibration. The collisional
broadening of the 7S-6P3/2 transition turns out to be slightly less than the corresponding values
found in the litterature for the 6S-6P3/2 transition : 7.2 [10] and 8.1x10-14 ncs [11].
Fig. 5.
-Damping rate of the 7S-6P3,2 coherence as a function of the Cs density. The dots and crosses were obtained in two different sealed glass cells.
Taking into account the spectral widths of the excitation line and the coherence damping
rate of the detection transition, one computes the natural half-width of the holes :
The difference with our experimental result (5.9 ± 1 MHz) sets an upper limit of 2 MHz for the instrumental broadening, possible origins of which are : residual frequency jitter of the lasers, residual Doppler width (due to imperfect alignment of the lasers), or traces of a foreign
gas in the sealed glass cell.
3.5.2 Evolution of Fp with ncs. - The height of the observed Doppler profile yields a value
for the damping rate r p of the global 6P3/2 population. This value was later confirmed by a
direct measurement of the time relaxation of the probe beam absorption after a pulsed
excitation of the forbidden transition.
In figure 6 the obtained values of Fp/2-r are plotted as a function of ncs. One sees
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a rapid decrease when ncs increases from 5 x 1012 to 5 x 1013 at/cm3 ;
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